I think this is the state of play.  If you use QBO you can't import invoices.  If you try you get the message "Import invoices isn't quite ready to support sales tax...Support...will be coming soon."  But QB support here is advising us to send a feature request to their product engineers, which means maybe not so soon.

Using a third party app, the workaround suggested by QB support, is brought up in the context of non-taxable invoicing.  

Using a third party app can you import taxable sales invoices into QBO?